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A Finite Element Recovery Approach to Eigenvalue Approximations with Applications to Electronic Structure Calculations

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Abstract

In this paper, we propose and analyze a recovery approach for trilinear finite element approximations on locally-refined hexahedral meshes for a class of elliptic eigenvalue problems. In the approach a local high-order interpolation recovery is followed by some gradient averaging based defect correction scheme. It is proved theoretically and shown numerically that our recovery approach can produce highly accurate eigenpair approximations. And we observe from our numerical experiments that the recovered eigenvalue approximation from the gradient averaging based defect correction approximates the exact eigenvalue from below. Furthermore, this approach has been applied to electronic structure calculations to improve the total energy approximations with small extra overheads.

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Acknowledgements

The authors would like to thank Prof. Xiaoying Dai, Prof. Xingao Gong, Prof. Lihua Shen, and Dr. Zhang Yang for their stimulating discussions and fruitful cooperations on electronic structure computations. The second author is grateful to Prof. Zeyao Mo for his encouragements. The authors would also like to thank the referees for their constructive comments and suggestions that improve the presentation of this paper.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jun Fang & Aihui Zhou

  2. Graduate University of Chinese Academy of Sciences, Beijing, 100190, China

    Jun Fang

  3. HPCC, Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China

    Xingyu Gao

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  1. Jun Fang
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  2. Xingyu Gao
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  3. Aihui Zhou
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Correspondence to Aihui Zhou.

Additional information

This work was partially supported by the National Science Foundation of China under grants 10871198, 10971059 and 61033009, the Funds for Creative Research Groups of China under grant 11021101, the National Basic Research Program of China under grants 2011CB309702 and 2011CB309703, and the National High Technology Research and Development Program of China under grant 2010AA012303.

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Fang, J., Gao, X. & Zhou, A. A Finite Element Recovery Approach to Eigenvalue Approximations with Applications to Electronic Structure Calculations. J Sci Comput 55, 432–454 (2013). https://doi.org/10.1007/s10915-012-9640-5

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  • DOI: https://doi.org/10.1007/s10915-012-9640-5

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